<p>This paper and its companion are concerned with the problems of 3-D object recognition and shape estimation from image curves using a 3-D object curve model that is invariant to affine transformation onto the image space, and a binocular stereo imaging system. The objects of interest here are the ones that have markings (e.g., characters, letters, special drawings and symbols, etc.) on their surfaces. The 3-D curves on the object are modeled as B-splines, which are characterized by a set of parameters (the control points) from which the 3-D curve can be totally generated. The B-splines are invariant under affine transformations. That means that the affine projected object curve onto the image space is a B-spline whose control points are related to the object control points through the affine transformation. Part I deals with issues relating to the curve modeling process. In particular, the authors address the problems of estimating the control points from the data curve, and of deciding on the "best" order B-spline and the "best" number of control points to be used to model the image or object curve(s). A minimum mean-square error (mmse) estimation technique which is invariant to affine transformations is presented as a noniterative, simple, and fast approach for control point estimation. The "best" B-spline is decided upon using a Bayesian selection rule. Finally, we present a matching algorithm that allocates a sample curve to one of p prototype curves when the sample curve is an a priori unknown affine transformation of one of the prototype curves stored in the data base. The approach is tried on a variety of images of real objects.</p>